> without a push system it is useless on modern mobile operating systems.

If by "push" you mean sending notifications through 3rd party servers like google's Firebase, I'd like not to use it.

I was using XMPP on mobile for years, receiving messages in background normal way almost with no battery drain.

I also run k9mail 24/7 with my personal IMAP server, receiving messages in realtime (imap idle) with almost no battery waste.

Battery is wasted more by bloated frameworks like Electron rather than by network activity itself.